In this talk, we show a global existence of the weak solution to the volume preserving mean curvature flow in the $d$-dimensional torus.

The weak solution is called a $L^2$-flow and is a family of Radon measures that satisfies an inequality such as Brakke's mean curvature flow.

The flow we obtained is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of ball with the same volume.

To construct the weak solution, we use the Allen-Cahn equation with non-local term motivated by studies of Mugnai, Seis, and Spadaro (2016), and Kim and Kwon (2020).

We show that the Allen-Cahn equation has the properties needed to construct the weak solution, such as the $L^2$-estimate of the Lagrange multiplier.